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Wheat on a Chess Board
© Copyright 2000, Jim Loy
There is a well-known story of the man who invented
chess. The local ruler was so pleased with the invention that he offered the
inventor a great reward in gold. The inventor suggested an alternative reward:
he would get one grain of wheat on the first square of the chess board, two
grains on the second square, four on the third, eight on the fourth, etc.,
doubling the number of grains each time. The ruler saw that this must be a much
better deal for him, and accepted. The board has 64 squares. How many total
grains of wheat did the ruler have to pay the inventor?
Answer: The number of grains on the 64th square is 2^63 (2 to the
63rd power). The total number of grains on the board is 2^64-1. These facts can
be easily deduced by considering just the first few squares, and generalizing
your findings. A proof can be done using mathematical induction, or geometric
series, or binary arithmetic. 2^64-1=18,446,744,073,709,551,615. That happens
to be much more wheat than exists in the whole world. In fact, that amount of
wheat would probably just fit in a building 25 miles long, 25 miles wide, and
1000 feet tall.
Addendum:
A similar puzzle is this: Take one sheet out of your newspaper, and fold it
in half, then fold it in half again, and again, and again. Can you fold it 30
times? Pretending that you can (you probably can't fold it more than eight
times), how thick would it be after 30 times? Assume the paper is 1/500"
thick or 1/200 cm. thick.
Answer: The newspaper, folded 30 times is about 34 mi. (54 km.)
thick. A few more folds and you could reach the moon!
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